Given a stream of numbers, generate a random number from the stream. You are allowed to use only O(1) space and the input is in the form of a stream, so can’t store the previously seen numbers.
So how do we generate a random number from the whole stream such that the probability of picking any number is 1/n. with O(1) extra space? This problem is a variation of Reservoir Sampling. Here the value of k is 1.
1) Initialize ‘count’ as 0, ‘count’ is used to store count of numbers seen so far in stream.
2) For each number ‘x’ from stream, do following
…..a) Increment ‘count’ by 1.
…..b) If count is 1, set result as x, and return result.
…..c) Generate a random number from 0 to ‘count-1’. Let the generated random number be i.
…..d) If i is equal to ‘count – 1’, update the result as x.
Random number from first 1 numbers is 1 Random number from first 2 numbers is 1 Random number from first 3 numbers is 3 Random number from first 4 numbers is 4
Auxiliary Space: O(1)
How does this work
We need to prove that every element is picked with 1/n probability where n is the number of items seen so far. For every new stream item x, we pick a random number from 0 to ‘count -1’, if the picked number is ‘count-1’, we replace the previous result with x.
To simplify proof, let us first consider the last element, the last element replaces the previously-stored result with 1/n probability. So the probability of getting the last element as the result is 1/n.
Let us now talk about the second last element. When the second last element processed the first time, the probability that it replaced the previous result is 1/(n-1). The probability that the previous result stays when the nth item is considered is (n-1)/n. So the probability that the second last element is picked in the last iteration is [1/(n-1)] * [(n-1)/n] which is 1/n.
Similarly, we can prove for third last element and others.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.
- Select a Random Node from a Singly Linked List
- Select a Random Node from a tree with equal probability
- Implement random-0-6-Generator using the given random-0-1-Generator
- Number of ways to select exactly K even numbers from given Array
- Random number generator in arbitrary probability distribution fashion
- Pseudo Random Number Generator (PRNG)
- Generating random number in a range in C
- Probability of getting a perfect square when a random number is chosen in a given range
- Select K elements from an array whose maximum value is minimized
- To check a number is palindrome or not without using any extra space
- Space efficient iterative method to Fibonacci number
- Number of n digit stepping numbers | Space optimized solution
- Random list of M non-negative integers whose sum is N
- Binomial Random Variables
- Erdos Renyl Model (for generating Random Graphs)
- Random Walk (Implementation in Python)
- Generating Random String Using PHP
- Probability of getting two consecutive heads after choosing a random coin among two different types of coins
- Generate a random permutation of 1 to N
- Generate a random permutation of elements from range [L, R] (Divide and Conquer)